We study the behavior of A-orbits in $G∕\mathrm{\Gamma }$ when G is a semisimple real algebraic $\mathbb{Q}$-group, Γ is a nonuniform arithmetic lattice, and A is a subgroup of the real torus of dimension $\ge {rank}_{\mathbb{Q}}G$. We show that every divergent trajectory of A diverges due to a purely algebraic reason, solving a long-standing conjecture of Weiss. In addition, we examine sets that intersect all A-orbits, and show that in many cases every A-orbit intersects every deformation retract $X\subset G∕\mathrm{\Gamma }$. This solves the questions raised by Pettet and Souto in 2009. The proofs use algebraic and differential topology, as well as algebraic group theory.

For any matroid M, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^{\bullet }(M)$ arising from hypersimplices. Using the mixed Hodge–Riemann relations, we deduce a strengthening of the log-concavity of the h-vector of a matroid complex, improving on an old conjecture of Dawson.

We study singular Kähler–Einstein metrics that are obtained as noncollapsed limits of polarized Kähler–Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the singularity, then the metric converges to the metric on its tangent cone at a polynomial rate on the level of Kähler potentials. When the tangent cone at the point has a smooth cross section, then the result implies polynomial convergence of the metric in the usual sense, generalizing a result due to Hein and Sun. We show that a similar result holds even in certain cases where the tangent cone is not locally isomorphic to the germ of the singularity. Finally, we prove a rigidity result for complete $\partial \overline{\partial }$-exact Calabi–Yau metrics with maximal volume growth. This generalizes a result of Conlon and Hein, which applies to the case of asymptotically conical manifolds.

Let $0\le s\le 1$ and $0\le t\le 2$. An $(s,t)$-Furstenberg set is a set $K\subset {\mathbb{R}}^2$ with the following property: there exists a line set $\mathcal{L}$ of Hausdorff dimension ${dim}_{\mathrm{H}}\mathcal{L}\ge t$ such that ${dim}_{\mathrm{H}}(K\cap \ell )\ge s$ for all $\ell \in \mathcal{L}$. We prove that for $s\in (0,1)$ and $t\in (s,2]$, the Hausdorff dimension of $(s,t)$-Furstenberg sets in ${\mathbb{R}}^2$ is no smaller than $2s+\mathit{ϵ}$, where $\mathit{ϵ}>0$ depends only on s and t. For $s>1∕2$ and $t=1$, this is an ϵ-improvement over a result of Wolff from 1999.

The same method also yields an ϵ-improvement to Kaufman’s projection theorem from 1968. We show that if $s\in (0,1)$, $t\in (s,2]$, and $K\subset {\mathbb{R}}^2$ is an analytic set with ${dim}_{\mathrm{H}}K=t$, then

$${dim}_{\mathrm{H}}\{e\in S^1:{dim}_{\mathrm{H}}{\mathrm{\pi }}_e(K)\le s\}\le s-\mathit{ϵ},$$

where $\mathit{ϵ}>0$ depends only on s and t. Here ${\mathrm{\pi }}_e$ is the orthogonal projection to the line in direction e.